That's why it came as a surprise to me when I was doing an Internet search and I ran across reference to National Bank Notes. This was American currency issued by local banks with a federal charter. From 1863 to 1935, local banks could deposit bonds in the US Treasury and then they could issue currency up to 90% of the value of the bonds. The banknotes were backed by the United States Government and often resembled United States Notes and later Federal Reserve Notes.
The cool thing about these National Bank Notes however, is that the name of the local bank is prominently displayed, meaning that your local town name was found on national currency. I quickly did a look and found that a bank in my hometown of San Luis Obispo, California was one of the towns issuing currency.
$10 San Luis Obispo, California, National Bank Note
Since the program continued all the way until 1935, the later smaller notes closely resemble the money we still have in circulation today.
$10 Paso Robles, California, National Bank Note
The $10 bill above is from the town where I teach which at the time it was issued probably had less than 3,000 people!